Modeling of light coupling effect using tunneling theorybased on particle properties of light

Ling-Feng Mao1• Jue Wang2

• H. Ning1• Changjun Hu1

•

Gaofeng Wang3• Mohammed M. Shabat4

Received: 15 January 2017 / Accepted: 21 September 2017 / Published online: 26 September 2017� Springer Science+Business Media, LLC 2017

Abstract Based on the wave-particle duality of light, the Schrodinger equation for a

photon as a particle is established to treat the light coupling effect by introducing concepts

of the virtual mass and potential for a photon, which is different from the previous method

that uses the analogy with quantum mechanics. The virtual mass is physically correlated to

the kinetic energy according to Einstein’s energy–momentum relation. As a consequence,

this new model has merits of physical simplicity and analytic nature. This new model can

well explain the exponential dependence of the light coupling effect on the physical

parameters in coupled waveguides on, which can be observed in the experimental and

simulation data reported in the literature. Moreover, an explicit expression for the coupling

length (coefficient) on the effects of physical parameters can be obtained by virtue of this

new model, whereas the previous modal approach and the coupled-wave model resulted in

implicit expressions. This new model does not only give a better physical understanding

but also offers a possibility to design and fabricate optic devices based on the light

coupling effect by optimizing the physical parameters.

Keywords Coupling � Tunneling � Waveguide � Wave-particle duality

& Ling-Feng Maomail_lingfeng@aliyun.com

& Jue Wangwangjue@sccas.cn

& Gaofeng Wanggaofeng@hdu.edu.cn

1 School of Computer and Communication Engineering, University of Science and TechnologyBeijing, Beijing 100083, China

2 Computer Network Information Center of CAS, Beijing 100190, China

3 School of Electronics and Information, Hangzhou Dianzi University, Hangzhou 310018, China

4 Physics Department, Islamic University of Gaza (IUG, Gaza), P.O. Box 108, Gaza Strip, Palestine

123

Opt Quant Electron (2017) 49:335DOI 10.1007/s11082-017-1174-5

1 Introduction

Coupling effect in waveguides (e.g., the evanescent coupling between two parallel optical

waveguides), which enables the transfer of light from one waveguide to an adjacent

waveguide, is very important for a variety of applications such as switching, power

division, dispersion compensation, and frequency (polarization) selection (Zheng et al.

2015; Moghaddam et al. 2010; Kuchinsky et al. 2002; Li et al. 2010; Wang et al. 2010;

Kumar et al. 1985; Mortensen et al. 2003; Uthayakumar et al. 2012; Obayya et al. 2000;

Singh 1999; Gorajoobi et al. 2013; Chaudhari and Gautam 2000; Headley et al. 2004;

Suzuki et al. 1994; Song et al. 2011; Sha et al. 2015; Henry and Verbeek 1989; Imotijevic

et al. 2007; Yu et al. 2014; Xu et al. 2014; Marom et al. 1984; Djordjev et al. 2002; Zhou

et al. 2013; Ali et al. 2008; Bird 1990; Shakouri et al. 1998; Sun 2007; Noda et al. 1981).

How the coupling length changes with the width of the separation layer on the coupling

structure was studied in Zheng et al. (2015), Li et al. (2010), Singh (1999), Gorajoobi et al.

(2013), Chaudhari and Gautam (2000), Suzuki et al. (1994), Song et al. (2011), Sha et al.

(2015), Imotijevic et al. (2007), Marom et al. (1984), Yariv (1973). The relation between

the coupling length and the wavelength was reported in Moghaddam et al. (2010), Kumar

et al. (1985), Mortensen et al. (2003), Uthayakumar et al. (2012), Gorajoobi et al. (2013),

Henry and Verbeek (1989), whereas the relation between the coupling length and the

relative dielectric constant of the separation layer was reported in Wang et al. (2010),

Obayya et al. (2000), Sun (2007), Noda et al. (1981), and the relation between the coupling

length and the width of the waveguide was reported in Yu et al. (2014), Xu et al. (2014).

Moreover, the relation between the coupling coefficient and the width of the separation

layer was studied in Li et al. (2010), Djordjev et al. (2002), Zhou et al. (2013), Ali et al.

(2008), Bird (1990), while the relation between the coupling coefficient and the frequency

(wavelength) of light was investigated in Moghaddam et al. (2010), Shakouri et al. (1998).

Control of the coupling length is certainly one deciding factor to the success of optic

devices based on directional couplers. Note that the method to obtain the coupling length is

approximate because there is not an exact analytical solution to the wave equation (Kumar

et al. 1985). The modal approach (Yariv 1973) and the coupled-wave approach (Suematsu

and Kishino 1977) are two approaches to the problem of two coupled waveguides. In

addition, an approach using the eigenvalue caused by the linear defect in the photonic

crystal was proposed to determine the coupling length (Kuchinsky et al. 2002). One

drawback of these approaches is that the physical origin of the coupling effect is unclear

and there is not exact analytical solution, although their results are consistent with the

experiments and simulations. A better understanding on the coupling effect can lead to

optical directional couplers, which are important building blocks in photonic circuits, with

better performances but simpler fabrication processes.

In studies of photonic circuits and optical devices, most researchers treat the optical

problem or the electromagnetic problem using a close mathematical analogy between the

Schrodinger equation and the Helmholtz equation for electromagnetic wave (Joannopoulos

et al. 2011; Olkhovsky 2014; Keller 2001; Lenstra and van Haeringen 1990; Dragoman and

Dragoman 2004 and the references therein). Over several decades of work, researchers

have only recently begun to understand the true nature of such a wave function for a photon

(Smith and Raymer 2007 and the references therein). There is a little study on photonic

circuits and optical devices via using the concept of the particle properties of light. In this

work, the particle properties of light are adopted to establish the Schrodinger equation for a

photon. Note that the particle tunneling can be also treated as an evanescent wave coupling

effect that occurs in quantum mechanics, while an evanescent wave coupling in optics

335 Page 2 of 18 L.-F. Mao et al.

123

refers to the coupling between two waveguides. Both the particle tunneling theory and the

uncertainty principle are utilized to explain the light coupling effect in this paper.

The wave-particle duality holds that matter and light exhibit properties of both waves

and particles. The purpose of this work is to shed some light on the coupling effect by

virtue of the tunneling theory and the uncertainty principle in quantum mechanics based on

the particle properties of light, which is different from the previous method that uses a

close mathematical analogy between the Schrodinger equation and the Helmholtz equation

to treat the optic problem or the electromagnetic problems. In particular, a new physical

model relevant to the coupling effect in coupled waveguides according to the uncertainty

principle and the tunneling theory is proposed. Through its physical simplicity and analytic

nature, such a new model can be applied to describe the experimental results and provide

the knowledge on how the physical parameters of coupled waveguides can affect the

performances.

This new model illustrates that the physical origin of the light coupling effect can be

explainable by the particle tunneling theory and the uncertainty principle because light is

also a particle. In addition, a concept of potential can be developed to measure the effective

dielectric constant for materials using the coupling effect of light. This new model also

states that the coupling length and the coupling efficient can be modulated or tuned by

choosing or modulating the width of waveguide, the separation between the waveguides,

and the effective dielectric constant of the separation. Hence, optical directional couplers

can be reasonably designed to achieve desirable performance by virtue of this new model.

One may find that this new model is not only physical simplicity but also has great value in

practical applications.

2 Theory

Electromagnetic field or light in a source-free dielectric space is described by Maxwell’s

equations (e.g., Fitzpatrick 2008)

r � D*

¼ q ¼ 0 ð1Þ

r � B*

¼ 0 ð2Þ

r � E*

¼ � oB

*

otð3Þ

r � B*

¼ leo E

*� �

ot¼ 1

c2

o E

*� �

otð4Þ

where q is the free charge density, l is the permittivity of the dielectric space, e is the

permeability of the dielectric space, c is the light velocity in the dielectric space, t is the

time, D*

is the displacement field vector, B*

is the magnetic field vector, and E*

is the electric

field vector. Noting that r�r� E*

¼ � 1�c2

� �o2E

*.ot2

� �and r�r� E

*

¼

r r � E*

� �� r2E

*

, we get

Modeling of light coupling effect using tunneling theory… Page 3 of 18 335

123

r2E*

� 1

c2

o2E

*

ot2¼ 0 ð5Þ

For a plane wave, E*

x; tð Þ ¼ E*

0ei k

*

�x*

�xt

� �, propagating in the x-direction in a lossless

medium space, Eq. (5) becomes

o2

ox2� 1

c2

o2

ot2

� �E*

¼ k2 � x2

c2

� �E*

0ei k

*

�x*

�xt

� �¼ 0 ð6Þ

where k*

is the wave vector, and x is the angular frequency. Equation (6) requires

k2 � x2�c2

� �¼ 0. Considering the photon energy of E ¼ hv ¼ �hx and its momentum of

p ¼ h=k ¼ �hk (herein h is the Planck constant, �h is the reduced Planck constant, and k is

the wavelength of light), Eq. (6) can be rewritten as

p2 � E2

c2

� �E*

0ei k

*

�x*

�xt

� �¼ 0 ð7Þ

The Schrodinger equation is used to describe a physical system with quantum effects

(such as wave-particle duality) being significant. There are two key parameters in the

Schrodinger equation. One is the effective mass of the particle, the other is the potential.

Thus, a virtual mass and the potential for a photon is need to be introduced for building a

Schrodinger equation of a photon. Such a virtual mass and the potential can be introduced

by comparing the Maxwell’s equations for a photon and the assumption that the potential is

zero when a photon travels in vacuum. Note that the photon momentum is determined by

the wavelength, it is correlated to the medium where a photon travels. Usually, the mass

does not change with the potential. Therefore, a virtual mass of a photon is determined by

the frequency of the light and independent of the medium where a photo travels. On the

other hand, the potential should be directly dependent of the medium where a photo travels.

When a virtual mass and the potential for a photon have been given, a Schrodinger

equation of a photon can be built, And thus we can totally copy the concept of tunneling

and the methods how to determine the tunneling in quantum mechanics to study the

coupling effect in waveguides via using the tunneling of a photon. Detail discuss will be

found in the following.

2.1 A Schrodinger equation for a photon

According to the wave-particle duality of light, the light is both a particle and a wave. Thus

the coupling effect in waveguides is similar to the case of a particle tunneling in double

well potential when the light is treated as a particle. Then we can study the photon

tunneling in the light of the methods used in the literature on tunneling in double well

potential (e.g., Song 2008; Verguilla-Berdecia 1993; Rastelli 2012; Khuat-duy and

Leboeuf 1993; Weiss et al. 1987; Benderskii and Kats 2002; Gangopadhyaya et al. 1993;

Park et al. 1998; Jelic and Marsiglio 2012; Kierig et al. 2008).

Assume that a photon can be treated as a non-relativistic particle moving along the x-

direction, and a particle in the free space can be described as a plane wave w x; tð Þ ¼Ae

i�h px�Etð Þ with E ¼ p2

2mþ V .where m is the effective mass of a particle, and V is the

potential of the particle (e.g., Shankar 2012). Note that such an effective mass is a virtual

335 Page 4 of 18 L.-F. Mao et al.

123

mass not a realistic mass for a photon. In order to build a wave equation based on the

particle properties of the wave-particle duality of light, such a virtual mass (m) for a photon

is introduced into the Schrodinger equation similar to the effective electron mass, which is

used to characterize the kinetic energy of a photon when it is treated as a particle (In the

following, we will discuss it in detail). Therefore, one obtains (e.g., Shankar 2012)

i�how x; tð Þ

ot¼ � �h2

2m

o2

ox2þ V

� �w x; tð Þ ð8Þ

Because such a ‘‘particle’’ is a photon, it is thus also a wave. At the same time, one can

also obtain

pc ¼ E ¼ p2

2mþ V ð9Þ

2.2 Potential for a photon

When the photon momentum p ¼ h=k ¼ �hk is the largest, i.e., the particle moves the

fastest, the corresponding wavelength is the shortest and the potential is the lowest (in the

valleys). It is well-known that the photon moves the fastest in the vacuum. Therefore, the

potential in the vacuum for a photon is defined as zero. According to Eq. (9), one obtains

m ¼ p

2c0

¼ �hx

2 c0ð Þ2ð10Þ

where c0 is the light velocity in the vacuum. According to Einstein’s energy–momentum

relation, Etot ¼ E2K þ m0c

2(Here Etot is total energy of a particle, EK is the sum of the

kinetic energy and the potential energy, m0 is the rest mass of a photon). Note that the rest

mass of a photon is zero, thus pc ¼ Etot ¼ 12mc2.we can also obtain the same expression as

Eq. 10. Note that Etot ¼ pcð Þ2þm0c2 ¼ E2

K þ m0c2, the Lorentz invariance is �m0c

2 the if

the system is static for a photon (which means that the momentum is zero).

When a photon is moving in a dielectric space, the velocity is less than the light velocity

in the vacuum, and thus it can be assumed that such a decrease in the velocity is caused by

the potential. Therefore, according to Eq. (9), the potential for a photon in a dielectric

material can be obtained as

V ¼ pc� p2

2m¼ �hx 1 � le

l0e0

� �ð11Þ

where l0 and e0 are the permittivity and permeability of the vacuum, respectively.

Considering that the potential in the vacuum is the lowest and defined as zero, a photon

behaves like a hole when el C e0l0, and an electron when el B e0l0. Hence, when

el C e0l0, the barrier height / seen by a photon moving in a dielectric space is given as

u ¼ �hxlel0e0

ð12Þ

Modeling of light coupling effect using tunneling theory… Page 5 of 18 335

123

2.3 Hole-like carrier tunneling

A photon moving through a dielectric material can be treated as a hole-like carrier tun-

neling. The time-independent one-dimensional Schrodinger equation for a photon moving

through a dielectric material with el C e0l0 can be written as (e.g., Shankar 2012)

r2 þ 2m

�h2E

w xð Þ ¼ 0 x\ 0 or x[ b ð13Þ

r2 þ 2m

�h2E � uð Þ

w xð Þ ¼ 0 0� x� b ð14Þ

where b denotes the thickness of the dielectric material. Using the usual plane wave

solution method, the wave functions in three regions can be found in Shankar (2012) and

the ‘‘Appendix’’. Therefore, the tunneling coefficient is given as

TC ¼ 2kkBð Þ2

2kkBð Þ2þ k2 þ k2Bð Þ2

sinh2 kBbð Þð15Þ

If kBb � 1 (such a condition can be satisfied when b� k or b � k), sinh kBbð Þ ! ekBb=2

and Eq. (19) becomes

TC ! 2kkBð Þ2

k2 þ k2Bð Þ2

ekBb=2ð Þ2! 4kkB

k2 þ k2B

� �2

e�2kBb ! 1

x2lee�2x

ffiffiffiffile

pb ð16Þ

2.4 Photon tunneling time

The tunneling time is the time that a particle takes to transverse a spatial region. The

concept of the tunneling time in related research has remained controversial over the past

80 years (Stuchebrukhov 2016; Yang and Guo 2016; Kullie 2016; Winful 2006; Buttiker

and Landauer 1982; Steinberg 1995 and the references therein). Previous studies have

demonstrated that the physics of tunneling is essentially identical for classical light waves

and quantum mechanical waves (Steinberg 1995 and the references therein). In this study,

the transmission time can be described as Steinberg (1995)

sT ¼ m

�hk

1

T

Zb

0

dx AekBx þ Be�kBx� �

Ae�kBx þ BekBx� �

ð17Þ

According to the above equation and A=Bj j ¼ e2kBb, it can be concluded that the main

contribution to the tunneling time is the A2 term. Therefore, the above equation can be

approximated as

335 Page 6 of 18 L.-F. Mao et al.

123

sT � mb

�hk

ik kB � ikð Þ2e2kBb

kB kB þ ikð Þ2e�kBb � kB � ikð Þ2

ekBbh i

������������

¼ mb

�hk

k kB � ikð Þ2ekBb

kB kB þ ikð Þ2e�2kBb � kB � ikð Þ2

h i������

������ð18Þ

If kBb[ [ 1 (e.g., b� k or b � k), one can has: e�2kBb ¼ 0. Therefore, Eq. (22) can be

further approximated as

sT ¼ mb

�hkBekBb ! ekBb ð19Þ

2.5 Photon tunneling in double waveguides

In double waveguides with a separation layer, the wave functions in five regions can be

found in Shankar (2012) and the ‘‘Appendix’’. Matching the wave functions and their

derivatives at the boundaries, one can obtain the tunneling coefficient:

TC ¼ 4k2B

e2ikw1 � eikbð Þ ik � kBð Þ2�e2ikw2 ik þ kBð Þ2h i

������������e

�2kB 2w1þbð Þ ! e�2xffiffiffiffile

p2w1þbð Þ ð20Þ

where w1, w2, and b are the width of waveguide 1, the width of waveguide 2, and the

thickness of the separation layer, respectively.

Noting that the power transfer ratio between the two waveguides is defined as the

coupling coefficient, thus the coupling coefficient is equivalent to the tunneling coefficient.

If the photons are inputted into the left waveguide (e.g., waveguide 1) and let A2 = 1, the

transmission time in the separation region (w1 B x B w1 ? b) can be described as

(Steinberg 1995)

sT ¼ m

�hk

1

T

Zw1þb

w1

dx A3ekBx þ B3e

�kBx� �

A3e�kBx þ B3e

kBx� �

ð21Þ

Note that w1 [ k and B3=A3 ! e2kBw1 , thus kBw1 � 1 and B3 � A3. This implies that

the main contribution to the tunneling time is the B23 term, and thus Eq. (33) leads to

sT ¼ mb

�hk

B23

A4

��������

¼ mb

�hk�

ik þ kBð Þ2e2ikw1 � eikb� �

e�2ikw1 ik � kBð Þ2� ik þ kBð Þ2

����������ekB 4w1þbð Þ

! m

�hkex

ffiffiffiffile

p4w1þbð Þ

ð22Þ

Modeling of light coupling effect using tunneling theory… Page 7 of 18 335

123

2.6 Uncertainty principle

According to the uncertainty relations between position and momentum and between time

and energy, one has

Dx � Dp� �h

2ð23Þ

Dt � DE� �h

2ð24Þ

where Dx is the uncertainty in position, Dp is the uncertainty in momentum, DE is the

uncertainty in energy, and Dt is the uncertainty in time. Since E ¼ pc, one can obtain:

DE ¼ c � Dp and Dp� �h= 2c � Dtð Þ.Since one cannot assure which waveguide a photon is in during the time interval for the

photon to travel through the separation layer, thus the uncertainty in the time for the photon

can be obtained as Dt ¼ sT .

The uncertainty principle implies that the more accurate the momentum is decided (i.e.,

a smaller Dp), the less accurate the position is known (i.e., a larger Dx). Therefore, the

uncertainty in the space (the x-direction) can be obtained as

Dx ¼ �h

2Dp� c � sT / mc

�hkex

ffiffiffiffile

p4w1þbð Þ ð25Þ

The above equation illustrates that the uncertainty in the space (the x-direction) for the

photon is the maximum length in the waveguide where the photon can be found. The

coupling length is defined as the maximum length of the light transmission along the

waveguide (Zheng et al. 2015; Moghaddam et al. 2010; Kuchinsky et al. 2002; Li et al.

2010; Wang et al. 2010; Kumar et al. 1985; Mortensen et al. 2003; Uthayakumar et al.

2012; Obayya et al. 2000; Singh 1999; Gorajoobi et al. 2013; Chaudhari and Gautam 2000;

Headley et al. 2004; Suzuki et al. 1994; Song et al. 2011; Sha et al. 2015; Henry and

Verbeek 1989; Imotijevic et al. 2007; Yu et al. 2014; Xu et al. 2014; Marom et al. 1984;

Djordjev et al. 2002; Zhou et al. 2013; Ali et al. 2008; Bird 1990; Shakouri et al. 1998; Sun

2007; Noda et al. 1981), when the light travels from one waveguide to the adjacent

waveguide in coupled waveguides. The coupling length is also defined as the distance for

which the maximum guided power is transferred from one medium to the other (Zheng

et al. 2015; Moghaddam et al. 2010; Kuchinsky et al. 2002; Li et al. 2010; Wang et al.

2010; Kumar et al. 1985; Mortensen et al. 2003; Uthayakumar et al. 2012; Obayya et al.

2000; Singh 1999; Gorajoobi et al. 2013; Chaudhari and Gautam 2000; Headley et al.

2004; Suzuki et al. 1994; Song et al. 2011; Sha et al. 2015; Henry and Verbeek 1989;

Imotijevic et al. 2007; Yu et al. 2014; Xu et al. 2014; Marom et al. 1984; Djordjev et al.

2002; Zhou et al. 2013; Ali et al. 2008; Bird 1990; Shakouri et al. 1998; Sun 2007; Noda

et al. 1981). Therefore, the maximum length that the photon can propagate or the uncer-

tainty in the space (the x-direction) for the photon is actually the coupling length, that is,

LC ¼ c � sT .

2.7 Coupling length and coupling coefficient

The equations of the coupling length and the coupling coefficient obtained by using the

particle properties of light and the traditional methods are compared in below. According

335 Page 8 of 18 L.-F. Mao et al.

123

to Eqs. (32) and (37), the coupling length and the coupling coefficient based on the

uncertainty principle and the tunneling theory can be obtained as

LC ¼ mc

�hk

ik þ kBð Þ2e2ikw1 � eikb� �

e�2ikw1 ik � kBð Þ2� ik þ kBð Þ2

����������ekB 4w1þbð Þ

! mc

�hkex

ffiffiffiffile

p4w1þbð Þ

ð26Þ

P / 4k2B

e2ikw1 � eikbð Þ ik � kBð Þ2�e2ikw2 ik þ kBð Þ2h i

������������e

�2kB 2w1þbð Þ ! e�2xffiffiffiffile

p2w1þbð Þ ð27Þ

The coupling length using the modal approach is given as Yariv (1973)

LC ¼ p2j

ð28Þ

where the parameter j can be calculated by assuming that the propagation constant of each

eigenvalue is essentially the same as that of an unperturbed waveguide via the modal

approach and the coupled-mode theory. The parameter j can be approximated only if the

modes are rigorously evaluated. The parameter j satisfies (Marom et al. 1984; Yariv 1973)

�iba þ icð ÞE1 þ jE2 ¼ 0

�jE1 þ �ibb þ icð ÞE2 ¼ 0ð29Þ

where c1;2 ¼ ba þ bb ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4j2 þ ba � bbð Þ2

q 2, and ba and bb are the propagating

constants of two waveguides, respectively. The above equations have solutions only for the

case of identical waveguides (i.e., ba ¼ bb). The power ratio between the normal modes

(which is herein named as the coupling coefficient) is given as Marom et al. (1984)

P ¼1 þ 4j2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4j2þ ba�bbð Þ2p

þ bb�bað Þ� �2

1 þ 4K2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4j2þ ba�bbð Þ2

p� bb�bað Þ

� �2

ð30Þ

For optical directional couplers based on photonic crystal, the coupling length can be

determined by Kuchinsky et al. (2002)

LC ¼ 0:5Dxx

V 0groupk0 ð31Þ

where Dx is the frequency splitting, V 0group is the group velocity in the unit of light velocity

in vacuum, and k0 is the light wavelength in vacuum. Since there is no analytical

expression for the frequency splitting, the coupling length can be determined only by

numerical calculations.

Note that the above equations start from the particle properties of light but the standard

quantization of the electromagnetic field or to the photon equation of for example Smith

and Raymer (2007) start from the wave properties. Thus, we can obtain a new physical

solution to the coupling effect based on the particle properties of light. Comparing with the

previous models (i.e., Eqs. 40, 43), this new model (i.e., Eqs. 38, 39) has merits of physical

simplicity and analytic nature. From this new model, one can readily find how the physical

Modeling of light coupling effect using tunneling theory… Page 9 of 18 335

123

parameters (e.g., the width of the waveguide, the width of the separation layer, and the

dielectric constant of the separation layer) affect the coupling length and the coupling

coefficient, whereas it is very difficult to directly find the effects of such physical

parameters on the coupling length and the coupling coefficient from the previous models.

Therefore, this new model is not only beautiful due to its physical simplicity but also has

great value in practical applications due to its analytic nature. In the following section, a

rich supply of data from nearly 30 published articles will be used to prove the validity of

this new model.

3 Results and discussion

Figure 1 depicts how the measured transmission phase time of light changes with the

barrier thickness. The experimental data is extracted from Fig. 3 in Balcou and Dutriaux

(1997). Herein, in all the figures, symbols denote the data from the literature, where lines

represent fitting curves to the data. A good exponential fit for the data of the measured

transmission phase time versus the barrier can be clearly observed in Fig. 1. Such an

exponential relation between the tunneling time and the barrier thickness is the same as

what this new model predicts (i.e., Eq. 23).

Note that there are oscillations in the curve for TE wave. Such an oscillation phe-

nomenon resulted from the quantum interference of material waves has been observed in

electrons tunneling through a barrier (Mao et al. 2000a, b, c, 2001a, b, c, 2002; Mao and

Wang 2008 and the references therein). Such oscillations for photons may have a similar

physical origin as that for electrons because both electron and light exhibit the wave-

particle duality. It is also found that such oscillations cannot be easily observed in

experiments because of the loss of the quantum interference of material waves (Mao and

Wang 2008). Similarly, it can be concluded that those oscillations are difficult to be

observed in experiments for light. The values of Adj. R2 are around 0.99 in the most cases

for the data fitting herein.

This new model (i.e., Eq. 38) predicts that the coupling length should exponentially

depend on the frequency of light, the width of the waveguide, the width of the separation

layer, the square root of the dielectric constant of the separation layer, and the permittivity

0 5 10 15 20 250

20

40

60

80

t(fs)

barrier width ( m)

Data: TM TEExponential fit:

Fig. 1 Impact of the barrierthickness on the transmissionphase time. The data is fromFig. 3 in Benderskii and Kats(2002)

335 Page 10 of 18 L.-F. Mao et al.

123

of the separation layer. Moreover, this new model (i.e., Eq. 39) also predicts that the

coupling coefficient has a similar behavior as the coupling length.

Figure 2 depicts the logarithm of the coupling length versus the width of the separation

layer. The data is from Zheng et al. (2015), Li et al. (2010), Singh (1999), Gorajoobi et al.

(2013), Chaudhari and Gautam (2000), Suzuki et al. (1994), Song et al. (2011), Sha et al.

(2015), Imotijevic et al. (2007), Marom et al. (1984), Yariv (1973). From Fig. 2, it can be

clearly observed that the logarithm of the coupling length linearly depends on the width of

the separation layer, which is the same as that predicted by this new model (i.e., Eq. 38).

The oscillations in the curve of the coupling length versus the width of the separation layer

can be observable only in the data from (Chaudhari and Gautam 2000), as shown in panel

(d). The physical origin of the oscillations is due to the quantum interference of photons

when the particle properties have been considered (Steinberg 1995; Balcou and Dutriaux

1997; Mao et al. 2000a, b, c, 2001a, b, c and the references therein).

Figure 3 depicts the coupling length versus the wavelength of light. The data is from

Moghaddam et al. (2010), Kumar et al. (1985), Mortensen et al. (2003), Uthayakumar et al.

(2012), Gorajoobi et al. (2013), Henry and Verbeek (1989). From Fig. 3, one can clearly

see that there is an exponential relation between the coupling length and the wavelength,

which is in good agreement with this new model (i.e., Eq. 38). Red lines in this figure and

the following figures represent the fitting curves using one-order exponential functions.

Figure 4 plots the coupling length versus the square root of the relative dielectric

constant of the separation layer. The data is from Wang et al. (2010), Obayya et al. (2000),

Sun (2007), Noda et al. (1981). One can readily observe that there is an exponential

120 160

1

2

2 3 4 5

0.4

0.8

200 400

-1012

2 3 4 5 6

3.4

3.6

3.8

0.2 0.4 0.6 0.8 1.0

3

4

1 2 3 4-0.5

0.0

0.5

400 600 800

1

2

8 9 10 11 12

0.0

0.4

0.8

100 200 300 400

1

2

12 13 140.4

0.8

1.2

(a) (b)

(c) (d)

(e)

Width of separation layer

ln(L

C)

(f)

(g) (h)

(i) (j)

Fig. 2 Impact of the width ofthe separation layer on thecoupling length. a Data fromFig. 6 in Zheng et al. (2015),b data from Fig. 5 in Singh(1999), c square symbols andcircle symbols denote data fromFig. 4 in Li et al. (2010) andFig. 2 in Gorajoobi et al. (2013)respectively, d data from Fig. 10in Chaudhari and Gautam (2000),e data from Fig. 2 in Suzuki et al.(1994), f data from Fig. 6 in Songet al. (2011), where square andcircle symbols denoteexperimental and simulation datarespectively, g data from Fig. 2in Sha et al. (2015), h data fromFig. 4 in Imotijevic et al. (2007),i data from Fig. 5 in Marom et al.(1984), and j data from Fig. 10 inYariv (1973). The separation is inthe unit of lm for (b) and(d) while in the unit of nm for allthe other cases

Modeling of light coupling effect using tunneling theory… Page 11 of 18 335

123

relation between the coupling length and the square root of the relative dielectric constant

of the separation layer, which is again in good agreement with this new model (i.e.,

Eq. 38). Small oscillations in the curve of the coupling length versus induced index dif-

ference can be observable only in the data from Obayya et al. (2000). All these oscillations

may have the same physical origin as that for electrons because the physics of tunneling for

0.6 1.2 1.80123456

10 20 30

102

103

104

0.5 1.0 1.50.0

0.4

0.8

1.2

6.0x10-4 6.3x10-4 6.6x10-4

5

10

15

(a)

0.02

0.04

0.224 0.228

L C

a/2 c0

(b)

(c)

L C

(d)

/

Fig. 3 Impact of the wavelength on the coupling length. a Square and circle symbols denote data fromFig. 2 in Kumar et al. (1985) and Fig. 2 in Uthayakumar et al. (2012) respectively, b square and circlesymbols denote data from Fig. 2 in Mortensen et al. (2003), and the data of the inset from Fig. 2 inMoghaddam et al. (2010) respectively, c data from Fig. 2 in Gorajoobi et al. (2013), and d data from Fig. 5in Henry and Verbeek (1989). All Adj. R2 values are around 0.99

9 10 1110

20

30

2.2992 2.2996

600

900

0

100

200

3.2 3.4 1.6 1.8

0.5

1.0

(b)

L C

(c)L C

1/2

(d)

(a)

Fig. 4 Impact of the relative dielectric constant of the separation layer on the coupling length. a Data fromFig. 2 in Wang et al. (2010), b data from Fig. 9 in Obayya et al. (2000), c data from Fig. 3 in Sun (2007),and d data from Fig. 4 in Noda et al. (1981)

335 Page 12 of 18 L.-F. Mao et al.

123

classical light waves is the same as that for quantum mechanical waves (Steinberg 1995,

and the references therein).

Figure 5 depicts how the coupling length changes with the width of the waveguide. The

data is from Yu et al. (2014), Xu et al. (2014). From Fig. 5, one can clearly see that there is

an exponential relation between the coupling length and the width of the waveguide, which

is in good agreement with this new model (i.e., Eq. 38). Figures 2, 3, 4 and 5 validate this

new model, which can well predict how the width of the separation layer, the wavelength

of light, the square root of the relative dielectric constant of the separation layer, and the

width of the waveguide affect the coupling length.

Figure 6 shows the coupling coefficient as a function of the width of the separation

layer. The data is from Li et al. (2010), Djordjev et al. (2002), Zhou et al. (2013). Ali et al.

(2008), Bird (1990). From Fig. 6, an exponential relation between the coupling coefficient

and the width of the separation layer can be observed, which is exactly the same as that

predicted by this new model (i.e., Eq. 39). Small oscillations in the curve of the coupling

coefficient versus the width of the separation layer can be observable only in the data from

Ali et al. (2008).

Figure 7 plots the coupling coefficient versus the frequency of light. The data is from

Moghaddam et al. (2010), Shakouri et al. (1998). From Fig. 7, it can be clearly observed

that there is an exponential relation between the coupling coefficient and the frequency

(wavelength) of light, which is again exactly as the same as that predicted by this new

model (i.e., Eq. 39). Obvious oscillations in the curve of the coupling coefficient versus the

frequency can be observable only in the data from Moghaddam et al. (2010). The reason

for such oscillations has been discussed in the preceding. Figures 6 and 7 validate this new

model, which can exactly predict how the width of the separation layer and the frequency

of light affect the coupling coefficient.

0.220 0.225 0.230

0.00

0.02

0.04

0.06

8.5 9.0 9.5 10.0

0.030

0.035

coup

ling

coef

ficie

nt

a/2 c0

(b)

frequency (GHz)

(a)Fig. 5 Impact of the waveguidewidth on the coupling length.a Data from Fig. 5 in Yu et al.(2014), and b data from Fig. 2 inXu et al. (2014)

Modeling of light coupling effect using tunneling theory… Page 13 of 18 335

123

4 Conclusions

A method for evaluating the coupling length and the coupling coefficient of parallel

coupled channel waveguides was proposed. This new model is based on the wave-particle

duality of light by virtue of the newly introduced concepts of the virtual mass and potential

for a photon, the uncertainty principle and the tunneling theory in quantum mechanics,

which is different from the previous method that uses a close mathematical analogy

between the Schrodinger equation and the Helmholtz equation to solve the optic problem

or the electromagnetic problem.

200 400

80

100

0 4 8 120

20

40

0

4

8

0.4 0.6 0.8 100 200 3000

4

8

(a)

coup

ling

coef

ficie

nt

Width of separation layer

(b)

(c) (d)

Fig. 6 Impact of the width of the separation layer on the coupling coefficient. a Data from Fig. 4 in Li et al.(2010), b square and circle symbols denote Fig. 5 in Djordjev et al. (2002) and Fig. 7 in Bird (1990), c datafrom Fig. 7 in Zhou et al. (2013), and d data from Fig. 3 in Ali et al. (2008)

0.00

0.02

0.04

0.06

0.220 0.225 0.230 8.5 9.0 9.5 10.0

0.030

0.035

coup

ling

coef

ficie

nt

a/2 c0

(b)

frequency (GHz)

(a)Fig. 7 Impact of the frequencyand wavelength on the couplingcoefficient. a Data from Fig. 2 inMoghaddam et al. (2010), andb data from Fig. 3 in Shakouriet al. (1998)

335 Page 14 of 18 L.-F. Mao et al.

123

From the particle properties of light (i.e., the light behaves both as a particle and as a

wave), the Schrodinger equation for a photon was firstly established via the assumption

that the largest photon momentum corresponds the fastest speed or the lowest potential,

and thus the potential in the vacuum is set as zero. Secondly, the law of the coupling

coefficient was deduced by solving the Schrodinger equation of a photon with assistance of

the particle tunneling theory in quantum mechanics. The particle properties of light show

that a photon through a dielectric layer behaves like a hole (when el C e0l0) or an electron

(when el B e0l0) through a barrier. Then, considering that the coupling length is caused by

the uncertainty in the space for the photon, the law of the coupling length was also deduced

by virtue of the uncertainty principle and the tunneling theory.

By analyzing the experimental (or simulation) data in nearly 30 previously published

articles, one can observe that the coupling length and the coupling coefficient are expo-

nentially dependent on the frequency of light, the width of the waveguide, the width of the

separation layer, and the square root of the dielectric constant of the separation layer,

which can be well predicted by this new model.

Acknowledgements This work was supported by the National Natural Science Foundation of China (underGrant No. 61774014).

Appendix

C Hole-like carrier tunneling: the wave functions in three regions: w xð Þ ¼ eikx þ Re�ikx

(x\ 0); w xð Þ ¼ Ae�kBx þ BekBx(0 B x B )b; w xð Þ ¼ Teikx(x[ b). Here k ¼ffiffiffiffiffiffiffi2mE

p

�h ,

kB ¼ffiffiffiffiffiffiffi2mu

p

�h ¼ xffiffiffiffiffile

p. Matching the wave functions and their derivatives at the boundaries

(x = 0 and x = b) leads to:1 þ R ¼ Aþ B, Ae�kBb þ BekBb ¼ Teikb, �kBAe�kBbþ

kBBekBb ¼ ikTeikb, and thus

T ¼ e�ikb 4ikkB

kB þ ikð Þ2e�kBb � kB � ikð Þ2

ekBbð32Þ

R ¼¼kBð Þ2þk2

h ie�kBb� kBð Þ2þk2

h iekBb

kB þ ikð Þ2e�kBb� kB � ikð Þ2

ekBbð33Þ

A ¼ 2ik kB � ikð ÞekBb

kB þ ikð Þ2e�kBb � kB � ikð Þ2

ekBbð34Þ

B ¼ 2ik kB þ ikð Þe�kBb

kB þ ikð Þ2e�kBb� kB � ikð Þ2

ekBbð35Þ

E Photon tunneling in double waveguides: the wave functions in five regions are

Region 1 ðx\0Þ : w xð Þ ¼ B1ekBx

Region 2 ð0� x�w1Þ : w xð Þ ¼ A2eikx þ B2e

�ikx

Region 3 ðw1 � x�w1 þ bÞ : w xð Þ ¼ A3e�kBx þ B3e

kBx

Modeling of light coupling effect using tunneling theory… Page 15 of 18 335

123

Region 4 ðw1 þ b\x\w1 þ w2 þ bÞ : w xð Þ ¼ A4e�kBx þ B4e

kBx

Region 5 ðx[w1 þ w2 þ bÞ : w xð Þ ¼ A5e�kBx

Matching the wave functions and their derivatives at the boundaries, one can obtain the

following equations:

B1

A2

¼ ik

ik þ kBð36Þ

B2

A2

¼ ik�kB

ik þ kBð37Þ

A3

A2

¼ 2kB ik þ kBð Þe�kBw1

k2B þ k2ð Þ eikw1�e�ikw1ð Þ ð38Þ

B3

A2

¼ 2kB ik þ kBð ÞekBw1

eikw1 ik þ kBð Þ2�e�ikw1 ik�kBð Þ2ð39Þ

A4

A3

¼ 2kB ik � kBð Þe�kB w1þbð Þ

eik w1þbð Þ e2ikw2 ik þ kBð Þ2� ik�kBð Þ2h i ð40Þ

B4

A4

¼ ik þ kBð Þik � kBð Þ e

2ik w1þw2þbð Þ ð41Þ

A4

A5

¼ e�kB w1þw2þbð Þ

eik w1þw2þbð Þ þ ikþkBð Þik�kBð Þ e

ik w1þw2þbð Þð42Þ

T ¼ A4

A2

¼ 4k2Be

�kB 2w1þbð Þ

e2ikw1�eikbð Þ ik�kBð Þ2�e2ikw2 ik þ kBð Þ2h i ð43Þ

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